In polymer chemistry, the molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species (Ni) and the molar mass (Mi) of that species. In linear polymers, the individual polymer chains rarely have exactly the same degree of polymerization and molar mass, and there is always a distribution around an average value. The molar mass distribution of a polymer may be modified by polymer fractionation.
Definitions of molar mass average Different average values can be defined, depending on the statistical method applied. In practice, four averages are used, representing the weighted mean taken with the mole fraction, the weight fraction, and two other functions which can be related to measured quantities:
Number average molar mass (Mn), also loosely referred to as number average molecular weight (NAMW). Mass average molar mass (Mw), where w stands for weight; also commonly referred to as weight average or weight average molecular weight (WAMW). Z-average molar mass (Mz), where z stands for centrifugation (from German Zentrifuge). Viscosity average molar mass (Mv).
M n = ∑ M i N i ∑ N i M w = ∑ M i 2 N i ∑ M i N i M z = ∑ M i 3 N i ∑ M i 2 N i M v = [ ∑ M i 1 + a N i ∑ M i N i ] 1 a {\displaystyle {\begin{aligned}M_{\mathrm {n} }&={\frac {\sum M_{i}N_{i}}{\sum N_{i}}}&&M_{\mathrm {w} }={\frac {\sum M_{i}^{2}N_{i}}{\sum M_{i}N_{i}}}\\M_{\mathrm {z} }&={\frac {\sum M_{i}^{3}N_{i}}{\sum M_{i}^{2}N_{i}}}&&M_{\mathrm {v} }=\left[{\frac {\sum M_{i}^{1+a}N_{i}}{\sum M_{i}N_{i}}}\right]^{\frac {1}{a}}\end{aligned}}}
Here, a is the exponent in the Mark–Houwink equation that relates the intrinsic viscosity to molar mass.
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